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Chapter 1: Problem 67

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.

Short Answer

Expert verified
The statement makes sense. The rectangular coordinate system indeed provides a geometric representation of an equation involving two variables, by plotting every pair of possible values as points in the system, creating a graphical representation of the equation.

Step by step solution

01

Understanding the Rectangular Coordinate System

A rectangular (or Cartesian) coordinate system is a pair of perpendicular lines (axes), usually horizontal and vertical, with a specific point (origin) identified as the intersection of these lines. In a two-dimensional space, any point can be located using the two coordinates which define the horizontal and vertical distances from the origin.
02

Linking Equations with Rectangular Coordinate System

An equation in two variables, let's say \(y = f(x)\), represents a relationship between the two variables \(x\) and \(y\). Every value of \(x\) corresponds to a value of \(y\) in this relationship. When we plot all such pairs of (x, y) on the rectangular coordinate system, we get a geometric picture of what an equation in two variables looks like.
03

Evaluating the Statement

Given the explanation above, the statement 'The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like' is correct. The rectangular coordinate system allows visualization of such equations, where each (x, y) pair from the equation is represented as a point in the system. Together, these points form a picture (like a line or curve) that represents the equation graphically.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations in Two Variables
An equation in two variables contains two different variables, typically represented as 'x' and 'y', and it shows a relationship between them. For example, a linear equation like \(y = 2x + 3\) tells us that for every value of 'x', there is a corresponding value of 'y' determined by doubling 'x' and adding three. Each pair of 'x' and 'y' values that satisfy the equation can be considered a solution to the equation.

When we work with these kinds of equations, we're not just looking at numbers; we are analyzing the relationship between these numbers. Understanding the connection between 'x' and 'y' helps us to solve problems, predict outcomes, and analyze situations in a variety of disciplines such as physics, economics, and engineering.
Cartesian Coordinate System
The Cartesian coordinate system, also known as the rectangular coordinate system, is the foundation of algebra and analytical geometry. Named after the French mathematician Ren茅 Descartes, it uses two axes鈥攈orizontal (x-axis) and vertical (y-axis)鈥攖hat intersect at a point called the origin (0,0).

The beauty of this system lies in its simplicity and functionality. It allows us to precisely plot any point in a two-dimensional space by using a pair of numerical coordinates: the first number (the x-coordinate) indicates the horizontal position, and the second number (the y-coordinate) shows the vertical position. It's like giving someone directions to a spot on a map using just two simple pieces of information.
Geometric Representation of Equations
The geometric representation of equations is a visual way to comprehend complex mathematical relations. When you have an equation in two variables, you can translate it into a visual form on a graph using the Cartesian coordinate system. Every solution to the equation鈥攅ach (x, y) pair鈥攃an be represented as a single point on the graph.

By plotting a set of points that are solutions to the equation, and then connecting these points, we can see the overall pattern that the relationship between 'x' and 'y' creates. This could be a line, a parabola, a circle, or any other shape, depending on the equation. This visualization helps us understand the behavior and properties of the equation, making it easier to solve and predict its results鈥攖ransforming abstract notions into tangible forms.
Plotting Points on a Graph
Plotting points on a graph is a practical application of the Cartesian coordinate system that brings equations to life. How do we plot a point? We start at the origin, move a certain distance along the x-axis (left or right based on whether the x-coordinate is negative or positive), then move in the direction of the y-axis (up or down for positive or negative y-coordinates).

For instance, the point (3, -2) means move 3 units to the right and 2 units down from the origin. Connect these individual points, and you get a visual representation or 'graph' of the equation. It's crucial for understanding functions, predicting trends, and even just visualizing relationships. Teaching yourself or students to plot points accurately is a vital skill in mastering mathematics.

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Most popular questions from this chapter

For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the ordered pairs (time of day, calories that I burned) to obtain a graph that is a horizontal line.

A contractor is to build a warehouse whose rectangular floor will have an area of 4000 square feet. The warehouse will be separated into two rectangular rooms by an interior wall. The cost of the exterior walls is \(\$ 175\) per linear foot and the cost of the interior wall is \(\$ 125\) per linear foot. Express the contractor's cost for building the walls, \(C,\) as a function of one of the dimensions of the warehouse's rectangular floor, \(x\).

Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries. $$\begin{array}{|l|c|cccccc|} \hline \text { Country } & \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} & \mathbf{E} & \mathbf{F} & \mathbf{G} \\\ \hline \begin{array}{l} \text { Liters of alcohol from } \\ \text { drinking wine, per } \\ \text { person per year }(x) \end{array} & 2.5 & 3.9 & 2.9 & 2.4 & 2.9 & 0.8 & 9.1 \\ \hline \begin{array}{l} \text { Deaths from heart } \\ \text { disease, per } 100,000 \\ \text { people per year }(y) \end{array} & 211 & 167 & 131 & 191 & 220 & 297 & 71 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|c|c|c|ccccc|c|c|} \hline \text { Country } & \mathbf{H} & \mathbf{I} & \mathbf{J} & \mathbf{K} & \mathbf{L} & \mathbf{M} & \mathbf{N} & \mathbf{O} & \mathbf{P} & \mathbf{Q} & \mathbf{R} & \mathbf{S} \\\ \hline(x) & 0.8 & 0.7 & 7.9 & 1.8 & 1.9 & 0.8 & 6.5 & 1.6 & 5.8 & 1.3 & 1.2 & 2.7 \\ \hline \mathbf{( y )} & 211 & 300 & 107 & 167 & 266 & 227 & 86 & 207 & 115 & 285 & 199 & 172 \\ \hline \end{array}$$ a. Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table. b. Use the scatter plot capability to draw a scatter plot of the data. c. Select the linear regression option. Use your utility to obtain values for \(a\) and \(b\) for the equation of the regression line, \(y=a x+b .\) You may also be given a correlation coefficient, \(r .\) Values of \(r\) close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of \(r\) close to \(-1\) indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of \(r\) close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data. d. Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} x^{2}+y^{2} &=9 \\ x-y &=3 \end{aligned}$$
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